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A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.

%I A277579 #24 Aug 01 2021 16:40:44
%S A277579 1,0,1,1,1,2,3,3,4,6,7,9,13,15,19,25,31,38,48,59,74,90,111,136,166,
%T A277579 201,246,297,357,431,522,621,745,892,1063,1263,1503,1780,2109,2491,
%U A277579 2941,3463,4077,4783,5616,6576,7689,8981,10486,12207,14209,16516,19178,22231
%N A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.
%C A277579 In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even terms of a finite sequence of positive integers.
%C A277579 For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
%C A277579 Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - _Gus Wiseman_, Jul 26 2021
%H A277579 Alois P. Heinz, <a href="/A277579/b277579.txt">Table of n, a(n) for n = 0..1000</a>
%e A277579 a(9) = 6: [2,1,1,1,1,1,1,1], [3,2,1,1,1,1], [3,3,2,1], [4,2,2,1], [4,3,1,1], [5,4].
%e A277579 a(10) = 7: [1,1,1,1,1,1,1,1,1,1], [3,2,2,1,1,1], [3,3,1,1,1,1], [4,2,1,1,1,1], [4,3,2,1], [5,5], [6,4].
%e A277579 a(11) = 9: [2,1,1,1,1,1,1,1,1,1], [3,2,1,1,1,1,1,1], [3,3,2,1,1,1], [3,3,3,2], [4,2,2,1,1,1], [4,3,1,1,1,1], [5,2,2,2], [5,4,1,1], [6,5].
%p A277579 with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = EP(P[k]) then c := c+1 else  end if end do: c end proc: seq(a(n), n = 0 .. 30);
%p A277579 n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = EP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
%p A277579 # alternative Maple program:
%p A277579 b:= proc(n, i, s, t) option remember; `if`(n=0,
%p A277579       `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
%p A277579       `if`(i>n, 0, b(n-i, i, s+t*i-irem(i+1, 2), -t))))
%p A277579     end:
%p A277579 a:= n-> b(n$2, 0, 1):
%p A277579 seq(a(n), n=0..60);
%t A277579 b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i+1, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 21 2016, translated from Maple *)
%t A277579 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]]; Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[conj[#],_?OddQ]&]],{n,0,15}] (* _Gus Wiseman_, Jul 26 2021 *)
%o A277579 (Sage)
%o A277579 def a(n):
%o A277579     AS = lambda s: abs(sum((-1)^i*t for i,t in enumerate(s)))
%o A277579     EP = lambda s: sum((t+1)%2 for t in s)
%o A277579     return sum(AS(p) == EP(p) for p in Partitions(n))
%o A277579 print([a(n) for n in (0..30)]) # _Peter Luschny_, Oct 21 2016
%Y A277579 The sign-sensitive version is A035457 (aerated version of A000009).
%Y A277579 Comparing odd parts to odd conjugate parts gives A277103.
%Y A277579 Comparing product of parts to product of conjugate parts gives A325039.
%Y A277579 Comparing the rev-alt sum to that of the conjugate gives A345196.
%Y A277579 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A277579 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A277579 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A277579 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A277579 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A277579 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A277579 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A277579 Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.
%K A277579 nonn
%O A277579 0,6
%A A277579 _Emeric Deutsch_ and _Alois P. Heinz_, Oct 20 2016